The fluid equations (REF )–() are spatially discretized
on a regular background grid of spacing \(h\)  so that \(\)  is divided into square or
cubic cells of side length \(h\) . Because of the checkerboard instability (see, e.g., [1]}) in solving the Navier-Stokes equations on collocated regular
grids, we use a MAC grid ([2]}) to stagger the components of Eulerian
vector quantities. For example, if \(\vec{v}\)  is an Eulerian vector-valued function,
\(\)  is a canonical basis vector, and \(\)  is a cell center, \(\cdot \vec{v}\)  is
discretized at \(- h/2\) . These discretization points comprise a new, regular
grid with spacing \(h\) . For the component under consideration, \(_h\)  denotes the set
of discretization points with cardinality \(n_\omega \) . The momentum equation (REF ) is discretized in time using a first-order method based on
the implicit-explicit Runge-Kutta method described in [3]}, and the
incompressibility condition () is satisfied using PmIII of [4]}.
